// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 20010-2011 Hauke Heibel <hauke.heibel@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_SPLINE_H
#define EIGEN_SPLINE_H

#include "SplineFwd.h"

namespace Eigen {
/**
     * \ingroup Splines_Module
     * \class Spline
     * \brief A class representing multi-dimensional spline curves.
     *
     * The class represents B-splines with non-uniform knot vectors. Each control
     * point of the B-spline is associated with a basis function
     * \f{align*}
     *   C(u) & = \sum_{i=0}^{n}N_{i,p}(u)P_i
     * \f}
     *
     * \tparam _Scalar The underlying data type (typically float or double)
     * \tparam _Dim The curve dimension (e.g. 2 or 3)
     * \tparam _Degree Per default set to Dynamic; could be set to the actual desired
     *                degree for optimization purposes (would result in stack allocation
     *                of several temporary variables).
     **/
template <typename _Scalar, int _Dim, int _Degree> class Spline
{
public:
    typedef _Scalar Scalar; /*!< The spline curve's scalar type. */
    enum
    {
        Dimension = _Dim /*!< The spline curve's dimension. */
    };
    enum
    {
        Degree = _Degree /*!< The spline curve's degree. */
    };

    /** \brief The point type the spline is representing. */
    typedef typename SplineTraits<Spline>::PointType PointType;

    /** \brief The data type used to store knot vectors. */
    typedef typename SplineTraits<Spline>::KnotVectorType KnotVectorType;

    /** \brief The data type used to store parameter vectors. */
    typedef typename SplineTraits<Spline>::ParameterVectorType ParameterVectorType;

    /** \brief The data type used to store non-zero basis functions. */
    typedef typename SplineTraits<Spline>::BasisVectorType BasisVectorType;

    /** \brief The data type used to store the values of the basis function derivatives. */
    typedef typename SplineTraits<Spline>::BasisDerivativeType BasisDerivativeType;

    /** \brief The data type representing the spline's control points. */
    typedef typename SplineTraits<Spline>::ControlPointVectorType ControlPointVectorType;

    /**
    * \brief Creates a (constant) zero spline.
    * For Splines with dynamic degree, the resulting degree will be 0.
    **/
    Spline() : m_knots(1, (Degree == Dynamic ? 2 : 2 * Degree + 2)), m_ctrls(ControlPointVectorType::Zero(Dimension, (Degree == Dynamic ? 1 : Degree + 1)))
    {
        // in theory this code can go to the initializer list but it will get pretty
        // much unreadable ...
        enum
        {
            MinDegree = (Degree == Dynamic ? 0 : Degree)
        };
        m_knots.template segment<MinDegree + 1>(0) = Array<Scalar, 1, MinDegree + 1>::Zero();
        m_knots.template segment<MinDegree + 1>(MinDegree + 1) = Array<Scalar, 1, MinDegree + 1>::Ones();
    }

    /**
    * \brief Creates a spline from a knot vector and control points.
    * \param knots The spline's knot vector.
    * \param ctrls The spline's control point vector.
    **/
    template <typename OtherVectorType, typename OtherArrayType>
    Spline(const OtherVectorType& knots, const OtherArrayType& ctrls) : m_knots(knots), m_ctrls(ctrls)
    {
    }

    /**
    * \brief Copy constructor for splines.
    * \param spline The input spline.
    **/
    template <int OtherDegree> Spline(const Spline<Scalar, Dimension, OtherDegree>& spline) : m_knots(spline.knots()), m_ctrls(spline.ctrls()) {}

    /**
     * \brief Returns the knots of the underlying spline.
     **/
    const KnotVectorType& knots() const { return m_knots; }

    /**
     * \brief Returns the ctrls of the underlying spline.
     **/
    const ControlPointVectorType& ctrls() const { return m_ctrls; }

    /**
     * \brief Returns the spline value at a given site \f$u\f$.
     *
     * The function returns
     * \f{align*}
     *   C(u) & = \sum_{i=0}^{n}N_{i,p}P_i
     * \f}
     *
     * \param u Parameter \f$u \in [0;1]\f$ at which the spline is evaluated.
     * \return The spline value at the given location \f$u\f$.
     **/
    PointType operator()(Scalar u) const;

    /**
     * \brief Evaluation of spline derivatives of up-to given order.
     *
     * The function returns
     * \f{align*}
     *   \frac{d^i}{du^i}C(u) & = \sum_{i=0}^{n} \frac{d^i}{du^i} N_{i,p}(u)P_i
     * \f}
     * for i ranging between 0 and order.
     *
     * \param u Parameter \f$u \in [0;1]\f$ at which the spline derivative is evaluated.
     * \param order The order up to which the derivatives are computed.
     **/
    typename SplineTraits<Spline>::DerivativeType derivatives(Scalar u, DenseIndex order) const;

    /**
     * \copydoc Spline::derivatives
     * Using the template version of this function is more efficieent since
     * temporary objects are allocated on the stack whenever this is possible.
     **/
    template <int DerivativeOrder>
    typename SplineTraits<Spline, DerivativeOrder>::DerivativeType derivatives(Scalar u, DenseIndex order = DerivativeOrder) const;

    /**
     * \brief Computes the non-zero basis functions at the given site.
     *
     * Splines have local support and a point from their image is defined
     * by exactly \f$p+1\f$ control points \f$P_i\f$ where \f$p\f$ is the
     * spline degree.
     *
     * This function computes the \f$p+1\f$ non-zero basis function values
     * for a given parameter value \f$u\f$. It returns
     * \f{align*}{
     *   N_{i,p}(u), \hdots, N_{i+p+1,p}(u)
     * \f}
     *
     * \param u Parameter \f$u \in [0;1]\f$ at which the non-zero basis functions 
     *          are computed.
     **/
    typename SplineTraits<Spline>::BasisVectorType basisFunctions(Scalar u) const;

    /**
     * \brief Computes the non-zero spline basis function derivatives up to given order.
     *
     * The function computes
     * \f{align*}{
     *   \frac{d^i}{du^i} N_{i,p}(u), \hdots, \frac{d^i}{du^i} N_{i+p+1,p}(u)
     * \f}
     * with i ranging from 0 up to the specified order.
     *
     * \param u Parameter \f$u \in [0;1]\f$ at which the non-zero basis function
     *          derivatives are computed.
     * \param order The order up to which the basis function derivatives are computes.
     **/
    typename SplineTraits<Spline>::BasisDerivativeType basisFunctionDerivatives(Scalar u, DenseIndex order) const;

    /**
     * \copydoc Spline::basisFunctionDerivatives
     * Using the template version of this function is more efficieent since
     * temporary objects are allocated on the stack whenever this is possible.
     **/
    template <int DerivativeOrder>
    typename SplineTraits<Spline, DerivativeOrder>::BasisDerivativeType basisFunctionDerivatives(Scalar u, DenseIndex order = DerivativeOrder) const;

    /**
     * \brief Returns the spline degree.
     **/
    DenseIndex degree() const;

    /** 
     * \brief Returns the span within the knot vector in which u is falling.
     * \param u The site for which the span is determined.
     **/
    DenseIndex span(Scalar u) const;

    /**
     * \brief Computes the span within the provided knot vector in which u is falling.
     **/
    static DenseIndex Span(typename SplineTraits<Spline>::Scalar u, DenseIndex degree, const typename SplineTraits<Spline>::KnotVectorType& knots);

    /**
     * \brief Returns the spline's non-zero basis functions.
     *
     * The function computes and returns
     * \f{align*}{
     *   N_{i,p}(u), \hdots, N_{i+p+1,p}(u)
     * \f}
     *
     * \param u The site at which the basis functions are computed.
     * \param degree The degree of the underlying spline.
     * \param knots The underlying spline's knot vector.
     **/
    static BasisVectorType BasisFunctions(Scalar u, DenseIndex degree, const KnotVectorType& knots);

    /**
     * \copydoc Spline::basisFunctionDerivatives
     * \param degree The degree of the underlying spline
     * \param knots The underlying spline's knot vector.
     **/
    static BasisDerivativeType BasisFunctionDerivatives(const Scalar u, const DenseIndex order, const DenseIndex degree, const KnotVectorType& knots);

private:
    KnotVectorType m_knots;         /*!< Knot vector. */
    ControlPointVectorType m_ctrls; /*!< Control points. */

    template <typename DerivativeType>
    static void BasisFunctionDerivativesImpl(const typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
                                             const DenseIndex order,
                                             const DenseIndex p,
                                             const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& U,
                                             DerivativeType& N_);
};

template <typename _Scalar, int _Dim, int _Degree>
DenseIndex Spline<_Scalar, _Dim, _Degree>::Span(typename SplineTraits<Spline<_Scalar, _Dim, _Degree>>::Scalar u,
                                                DenseIndex degree,
                                                const typename SplineTraits<Spline<_Scalar, _Dim, _Degree>>::KnotVectorType& knots)
{
    // Piegl & Tiller, "The NURBS Book", A2.1 (p. 68)
    if (u <= knots(0))
        return degree;
    const Scalar* pos = std::upper_bound(knots.data() + degree - 1, knots.data() + knots.size() - degree - 1, u);
    return static_cast<DenseIndex>(std::distance(knots.data(), pos) - 1);
}

template <typename _Scalar, int _Dim, int _Degree>
typename Spline<_Scalar, _Dim, _Degree>::BasisVectorType
Spline<_Scalar, _Dim, _Degree>::BasisFunctions(typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
                                               DenseIndex degree,
                                               const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& knots)
{
    const DenseIndex p = degree;
    const DenseIndex i = Spline::Span(u, degree, knots);

    const KnotVectorType& U = knots;

    BasisVectorType left(p + 1);
    left(0) = Scalar(0);
    BasisVectorType right(p + 1);
    right(0) = Scalar(0);

    VectorBlock<BasisVectorType, Degree>(left, 1, p) = u - VectorBlock<const KnotVectorType, Degree>(U, i + 1 - p, p).reverse();
    VectorBlock<BasisVectorType, Degree>(right, 1, p) = VectorBlock<const KnotVectorType, Degree>(U, i + 1, p) - u;

    BasisVectorType N(1, p + 1);
    N(0) = Scalar(1);
    for (DenseIndex j = 1; j <= p; ++j)
    {
        Scalar saved = Scalar(0);
        for (DenseIndex r = 0; r < j; r++)
        {
            const Scalar tmp = N(r) / (right(r + 1) + left(j - r));
            N[r] = saved + right(r + 1) * tmp;
            saved = left(j - r) * tmp;
        }
        N(j) = saved;
    }
    return N;
}

template <typename _Scalar, int _Dim, int _Degree> DenseIndex Spline<_Scalar, _Dim, _Degree>::degree() const
{
    if (_Degree == Dynamic)
        return m_knots.size() - m_ctrls.cols() - 1;
    else
        return _Degree;
}

template <typename _Scalar, int _Dim, int _Degree> DenseIndex Spline<_Scalar, _Dim, _Degree>::span(Scalar u) const
{
    return Spline::Span(u, degree(), knots());
}

template <typename _Scalar, int _Dim, int _Degree> typename Spline<_Scalar, _Dim, _Degree>::PointType Spline<_Scalar, _Dim, _Degree>::operator()(Scalar u) const
{
    enum
    {
        Order = SplineTraits<Spline>::OrderAtCompileTime
    };

    const DenseIndex span = this->span(u);
    const DenseIndex p = degree();
    const BasisVectorType basis_funcs = basisFunctions(u);

    const Replicate<BasisVectorType, Dimension, 1> ctrl_weights(basis_funcs);
    const Block<const ControlPointVectorType, Dimension, Order> ctrl_pts(ctrls(), 0, span - p, Dimension, p + 1);
    return (ctrl_weights * ctrl_pts).rowwise().sum();
}

/* --------------------------------------------------------------------------------------------- */

template <typename SplineType, typename DerivativeType>
void derivativesImpl(const SplineType& spline, typename SplineType::Scalar u, DenseIndex order, DerivativeType& der)
{
    enum
    {
        Dimension = SplineTraits<SplineType>::Dimension
    };
    enum
    {
        Order = SplineTraits<SplineType>::OrderAtCompileTime
    };
    enum
    {
        DerivativeOrder = DerivativeType::ColsAtCompileTime
    };

    typedef typename SplineTraits<SplineType>::ControlPointVectorType ControlPointVectorType;
    typedef typename SplineTraits<SplineType, DerivativeOrder>::BasisDerivativeType BasisDerivativeType;
    typedef typename BasisDerivativeType::ConstRowXpr BasisDerivativeRowXpr;

    const DenseIndex p = spline.degree();
    const DenseIndex span = spline.span(u);

    const DenseIndex n = (std::min)(p, order);

    der.resize(Dimension, n + 1);

    // Retrieve the basis function derivatives up to the desired order...
    const BasisDerivativeType basis_func_ders = spline.template basisFunctionDerivatives<DerivativeOrder>(u, n + 1);

    // ... and perform the linear combinations of the control points.
    for (DenseIndex der_order = 0; der_order < n + 1; ++der_order)
    {
        const Replicate<BasisDerivativeRowXpr, Dimension, 1> ctrl_weights(basis_func_ders.row(der_order));
        const Block<const ControlPointVectorType, Dimension, Order> ctrl_pts(spline.ctrls(), 0, span - p, Dimension, p + 1);
        der.col(der_order) = (ctrl_weights * ctrl_pts).rowwise().sum();
    }
}

template <typename _Scalar, int _Dim, int _Degree>
typename SplineTraits<Spline<_Scalar, _Dim, _Degree>>::DerivativeType Spline<_Scalar, _Dim, _Degree>::derivatives(Scalar u, DenseIndex order) const
{
    typename SplineTraits<Spline>::DerivativeType res;
    derivativesImpl(*this, u, order, res);
    return res;
}

template <typename _Scalar, int _Dim, int _Degree>
template <int DerivativeOrder>
typename SplineTraits<Spline<_Scalar, _Dim, _Degree>, DerivativeOrder>::DerivativeType Spline<_Scalar, _Dim, _Degree>::derivatives(Scalar u,
                                                                                                                                   DenseIndex order) const
{
    typename SplineTraits<Spline, DerivativeOrder>::DerivativeType res;
    derivativesImpl(*this, u, order, res);
    return res;
}

template <typename _Scalar, int _Dim, int _Degree>
typename SplineTraits<Spline<_Scalar, _Dim, _Degree>>::BasisVectorType Spline<_Scalar, _Dim, _Degree>::basisFunctions(Scalar u) const
{
    return Spline::BasisFunctions(u, degree(), knots());
}

/* --------------------------------------------------------------------------------------------- */

template <typename _Scalar, int _Dim, int _Degree>
template <typename DerivativeType>
void Spline<_Scalar, _Dim, _Degree>::BasisFunctionDerivativesImpl(const typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
                                                                  const DenseIndex order,
                                                                  const DenseIndex p,
                                                                  const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& U,
                                                                  DerivativeType& N_)
{
    typedef Spline<_Scalar, _Dim, _Degree> SplineType;
    enum
    {
        Order = SplineTraits<SplineType>::OrderAtCompileTime
    };

    const DenseIndex span = SplineType::Span(u, p, U);

    const DenseIndex n = (std::min)(p, order);

    N_.resize(n + 1, p + 1);

    BasisVectorType left = BasisVectorType::Zero(p + 1);
    BasisVectorType right = BasisVectorType::Zero(p + 1);

    Matrix<Scalar, Order, Order> ndu(p + 1, p + 1);

    Scalar saved, temp;  // FIXME These were double instead of Scalar. Was there a reason for that?

    ndu(0, 0) = 1.0;

    DenseIndex j;
    for (j = 1; j <= p; ++j)
    {
        left[j] = u - U[span + 1 - j];
        right[j] = U[span + j] - u;
        saved = 0.0;

        for (DenseIndex r = 0; r < j; ++r)
        {
            /* Lower triangle */
            ndu(j, r) = right[r + 1] + left[j - r];
            temp = ndu(r, j - 1) / ndu(j, r);
            /* Upper triangle */
            ndu(r, j) = static_cast<Scalar>(saved + right[r + 1] * temp);
            saved = left[j - r] * temp;
        }

        ndu(j, j) = static_cast<Scalar>(saved);
    }

    for (j = p; j >= 0; --j) N_(0, j) = ndu(j, p);

    // Compute the derivatives
    DerivativeType a(n + 1, p + 1);
    DenseIndex r = 0;
    for (; r <= p; ++r)
    {
        DenseIndex s1, s2;
        s1 = 0;
        s2 = 1;  // alternate rows in array a
        a(0, 0) = 1.0;

        // Compute the k-th derivative
        for (DenseIndex k = 1; k <= static_cast<DenseIndex>(n); ++k)
        {
            Scalar d = 0.0;
            DenseIndex rk, pk, j1, j2;
            rk = r - k;
            pk = p - k;

            if (r >= k)
            {
                a(s2, 0) = a(s1, 0) / ndu(pk + 1, rk);
                d = a(s2, 0) * ndu(rk, pk);
            }

            if (rk >= -1)
                j1 = 1;
            else
                j1 = -rk;

            if (r - 1 <= pk)
                j2 = k - 1;
            else
                j2 = p - r;

            for (j = j1; j <= j2; ++j)
            {
                a(s2, j) = (a(s1, j) - a(s1, j - 1)) / ndu(pk + 1, rk + j);
                d += a(s2, j) * ndu(rk + j, pk);
            }

            if (r <= pk)
            {
                a(s2, k) = -a(s1, k - 1) / ndu(pk + 1, r);
                d += a(s2, k) * ndu(r, pk);
            }

            N_(k, r) = static_cast<Scalar>(d);
            j = s1;
            s1 = s2;
            s2 = j;  // Switch rows
        }
    }

    /* Multiply through by the correct factors */
    /* (Eq. [2.9])                             */
    r = p;
    for (DenseIndex k = 1; k <= static_cast<DenseIndex>(n); ++k)
    {
        for (j = p; j >= 0; --j) N_(k, j) *= r;
        r *= p - k;
    }
}

template <typename _Scalar, int _Dim, int _Degree>
typename SplineTraits<Spline<_Scalar, _Dim, _Degree>>::BasisDerivativeType Spline<_Scalar, _Dim, _Degree>::basisFunctionDerivatives(Scalar u,
                                                                                                                                    DenseIndex order) const
{
    typename SplineTraits<Spline<_Scalar, _Dim, _Degree>>::BasisDerivativeType der;
    BasisFunctionDerivativesImpl(u, order, degree(), knots(), der);
    return der;
}

template <typename _Scalar, int _Dim, int _Degree>
template <int DerivativeOrder>
typename SplineTraits<Spline<_Scalar, _Dim, _Degree>, DerivativeOrder>::BasisDerivativeType
Spline<_Scalar, _Dim, _Degree>::basisFunctionDerivatives(Scalar u, DenseIndex order) const
{
    typename SplineTraits<Spline<_Scalar, _Dim, _Degree>, DerivativeOrder>::BasisDerivativeType der;
    BasisFunctionDerivativesImpl(u, order, degree(), knots(), der);
    return der;
}

template <typename _Scalar, int _Dim, int _Degree>
typename SplineTraits<Spline<_Scalar, _Dim, _Degree>>::BasisDerivativeType
Spline<_Scalar, _Dim, _Degree>::BasisFunctionDerivatives(const typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
                                                         const DenseIndex order,
                                                         const DenseIndex degree,
                                                         const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& knots)
{
    typename SplineTraits<Spline>::BasisDerivativeType der;
    BasisFunctionDerivativesImpl(u, order, degree, knots, der);
    return der;
}
}  // namespace Eigen

#endif  // EIGEN_SPLINE_H
